Symmetry Representations and Characters

In Chapters 1-3, the symmetry operations which lead to identical atomic arrangements for a wide range of point groups have been discussed. In a few cases, additional objects such as vectors on particular atom centres were used to highlight the differences between 

Molecular Symmetry David J. Willock 2009 John Wiley Sons, Ltd. ISBN 978-0 70-85347-4 

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It is clear that each of the p-orbitals on the O atom in the H2O example responds differently to the symmetry operations sometimes they are unaffected and sometimes they are reversed. Rather than draw out each orbital every time, it is possible to use a simple numerical scheme to represent the effect of each symmetry operation on each orbital. These numbers are known as the characters of the operation in the representation. 

The characters give numerical factors relating each transformed orbital to its original form. If it is unchanged by the operation we write 1 and if it is reversed we write -1. 

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